Geodesic
Martingale Rosenthal inequalities in symmetric spaces
Sbornik. Mathematics, Tome 205 (2014) no. 12, pp. 1720-1740 Cet article a éte moissonné depuis la source Math-Net.Ru

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We establish inequalities similar to the classical Rosenthal inequalities for sequences of martingale differences in general symmetric spaces; a central role is played here by the predictable quadratic characteristic of a martingale. Bibliography: 26 titles.
Keywords: martingale differences, Rosenthal's inequality, symmetric space, Boyd index.
Mots-clés : martingale 
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S. V. Astashkin. Martingale Rosenthal inequalities in symmetric spaces. Sbornik. Mathematics, Tome 205 (2014) no. 12, pp. 1720-1740. http://geodesic.mathdoc.fr/item/SM_2014_205_12_a2/

[1] H. P. Rosenthal, “On the subspaces of $L^p$ ($p>2$) spanned by sequences of independent random variables”, Israel J. Math., 8:3 (1970), 273–303 | DOI | MR | Zbl

[2] W. B. Johnson, G. Schechtman, “Sums of independent random variables in rearrangement invariant function spaces”, Ann. Probab., 17:2 (1989), 789–808 | DOI | MR | Zbl

[3] W. B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, “Symmetric structures in Banach spaces”, Mem. Amer. Math. Soc., 19, no. 217, 1979, v+298 pp. | DOI | MR | Zbl

[4] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, v. II, Ergeb. Math. Grenzgeb., 97, Function spaces, Springer-Verlag, Berlin–New York, 1979, x+243 pp. | MR | Zbl

[5] S. V. Astashkin, F. A. Sukochev, “Series of independent random variables in rearrangement invariant spaces: an operator approach”, Israel J. Math., 145:1 (2005), 125–156 | DOI | MR | Zbl

[6] S. V. Astashkin, F. A. Sukochev, “Series of independent, mean zero random variables in rearrangement-invariant spaces having the Kruglov property”, J. Math. Sci. (N. Y.), 148:6 (2008), 795–809 | DOI | MR

[7] S. V. Astashkin, F. A. Sukochev, “Independent functions and the geometry of Banach spaces”, Russian Math. Surveys, 65:6 (2010), 1003–1081 | DOI | DOI | MR | Zbl

[8] M. Sh. Braverman, Independent random variables and rearrangement invariant spaces, London Math. Soc. Lecture Note Ser., 194, Cambridge Univ. Press, Cambridge, 1994, viii+116 pp. | DOI | MR | Zbl

[9] V. M. Kruglov, “A note on infinitely divisible distributions”, Theory Probab. Appl., 15:2 (1970), 319–324 | DOI | MR | Zbl

[10] D. L. Burkholder, “Distribution function inequalities for martingales”, Ann. Probability, 1:1 (1973), 19–42 | DOI | MR | Zbl

[11] N. L. Bassily, “On two-sided inequalities for stopped random walks in Orlicz spaces”, Ann. Univ. Sci. Budapest Sect. Comput., 11 (1991), 63–73 | MR | Zbl

[12] G. Peškir, A. N. Shiryaev, “The Khintchine inequalities and martingale expanding sphere of their action”, Russian Math. Surveys, 50:5 (1995), 849–904 | DOI | MR | Zbl

[13] D. L. Burkholder, R. F. Gundy, “Extrapolation and interpolation of quasi-linear operators on martingales”, Acta Math., 124:1 (1970), 249–304 | DOI | MR | Zbl

[14] D. L. Burkholder, B. J. Davis, R. F. Gundy, “Integral inequalities for convex functions of operators on martingales”, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), v. II, Probability theory, Univ. California Press, 1972, 223–240 | MR | Zbl

[15] S. Astashkin, F. Sukochev, Chin Pin Wong, “Disjointification of martingale differences and conditionally independent random variables with some applications”, Studia Math., 205:2 (2011), 171–200 | DOI | MR | Zbl

[16] S. V. Astashkin, “Rosenthal type inequalities for martingales in symmetric spaces”, Russian Math. (Iz. VUZ), 56:11 (2012), 52–57 | DOI | MR | Zbl

[17] S. G. Kreĭn, Yu. I. Petunin, E. M. Semenov, Interpolation of linear operators, Transl. Math. Monogr., 54, Amer. Math. Soc., Providence, R.I., 1982, xii+375 pp. | MR | MR | Zbl | Zbl

[18] B. S. Kashin, A. A. Saakyan, Orthogonal series, Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, 1989, xii+451 pp. | MR | MR | Zbl | Zbl

[19] A. N. Shiryayev, Probability, Grad. Texts in Math., 95, Springer-Verlag, New York, 1984, xi+577 pp. | DOI | MR | MR | Zbl | Zbl

[20] R. J. Elliott, Stochastic calculus and applications, Appl. Math. (N. Y.), 18, Springer-Verlag, New York, 1982, ix+302 pp. | MR | MR | Zbl | Zbl

[21] J. Neveu, Discrete parameter martingales, North-Holland Math. Library, 10, North-Holland Publishing Co., Amsterdam–Oxford; American Elsevier Publishing Co., Inc., 1975, viii+236 pp. | MR | Zbl

[22] T. Shimogaki, “A note on norms of compression operators on function spaces”, Proc. Japan Akad., 46:3 (1970), 239–242 | DOI | MR | Zbl

[23] P. Hitczenko, S. Montgomery-Smith, “Measuring the magnitude of sums of independent random variables”, Ann. Probab., 29:1 (2001), 447–466 | DOI | MR | Zbl

[24] S. Kwapień, W. A. Woyczyński, Random series and stohastic integrals: single and multiple, Birkhäuser Boston, Inc., Probab. Appl., Boston, MA, 1992, xvi+360 pp. | MR | Zbl

[25] W. B. Johnson, G. Schechtman, “Martingale inequalities in rearrangement invariant function spaces”, Israel J. Math., 64:3 (1988), 267–275 | DOI | MR | Zbl

[26] M. Kikuchi, “On the Davis inequality in Banach function spaces”, Math. Nachr., 281:5 (2008), 697–709 | DOI | MR | Zbl